In traditional ellipsometry, the polarization state of a beam of electromagnetic radiation is modulated and/or analyzed by varying at least one polarization state parameter as a function of time. Rotating-optic ellipsometers linearly vary the azimuthal position of at least one optical element by rotating the element. Each rotating element induces a temporal modulation in the signal intensity at a frequency related to the rotation rate of the optic. Early developments employed rotating analyzers or polarizers which were unable to measure the sign of the phase change (Δ) caused by a sample. Later improvements took advantage of rotating compensators to provide sensitivity to this parameter. Another type of ellipsometer uses photoelastic modulators to vary the retardance of an optic as a function of time, typically in a sinusoidal manner by imparting a voltage on a piezoelectric transducer that induces stress in a photoelastic crystal. This stress results in a varying birefringence within the crystal and results in retardance in the measurement beam. In each of the above types of ellipsometer, the temporal modulation in the signal intensity is analyzed to determine optical properties of the sample.
In general, elements such as rotating compensators and photoelastic modulators have several drawbacks. Because the polarization modulation is a function of time, multiple measurement frames must be captured to completely describe the polarization state of the beam. For this reason, the measurement speed for any temporally modulated ellipsometer is fundamentally limited by its hardware. For moving or rapidly changing samples, such as in coating processes, it would be advantageous to complete a measurement with stationary optics. A stationary-element ellipsometer could also potentially be more stable, simple, and compact than a temporally modulated system.
To overcome the limitations of prior art ellipsometers, several designs for systems that eliminate the use of temporal modulation of the detected beam have been proposed. Channeled ellipsometers and polarimeters encode information about the polarization state of the beam onto the same dimension of the detector utilized for spectral or spatial information. In spectrally channeled ellipsometry, this is accomplished using a multi-order retarder. The retarder has polarization effects that are strongly wavelength-dependent, thereby creating a strong modulation pattern at a higher frequency within the spectral intensity profile. Similarly, spatially channeled systems modulate the intensity along one or both dimensions of an image plane by imaging a spatially varying optic such as a wedged retarder. Systems that utilize this approach have potential to capture a great deal of information about the spectral, polarization, and spatial components of a beam simultaneously, as described by Oka et al. in U.S. Pat. No. 7,336,360B2. The major drawback of channeled systems is that the information related to each variable must be separated. In much prior art, this is accomplished by Fourier transforming the resulting intensity information to pick out the characteristic frequencies of each individual source of modulation. Several drawbacks of this technology include more complex signal processing, increased noise, and decreased resolution.
Snapshot ellipsometers and polarimeters can provide the benefits of stationary optics without the drawbacks of channeled systems. In snapshot systems, spatial modulation of the signal is induced by imaging a spatially varying optical element onto a dedicated dimension of a multi-element detector. If a two-dimensional detector is used, the other dimension can be used to capture information related to spectral, angular, or spatial characteristics of a sample or beam. In the case of snapshot spectroscopic ellipsometry, an additional element separates the electromagnetic radiation spectrally along the orthogonal detector dimension, allowing full characterization of the polarization state and spectral profile of a beam in a single frame capture of the detector. Because the spectral separation and polarization modulation directions are independent, signal processing is straightforward and analogous to traditional ellipsometry techniques.
Mueller-Stokes calculus can be used to express the change in polarization caused by each element within the optical train of an ellipsometer. The polarization state of electromagnetic radiation is represented by a Stokes Vector, and each element is described by a Mueller Matrix that describes the polarization effects of the optic.
Mueller-Stokes descriptions for some common polarization elements are provided below.
Unpolarized light is characterized by intensity, I, and is described by the following Stokes vector:
      Unpolarized    ⁢                  ⁢          Light      ⁡              (        L        )              =      I    ⁡          (                                    1                                                0                                                0                                                0                              )      
The matrix below converts the Stokes vector representing a polarized beam to a scalar representation of the intensity on the detector.Detector(D)=Attn·(1 0 0 0)
A polarizer is an element with a characteristic pass axis that transmits only electromagnetic radiation with a polarization oriented along said axis.
      Polarizer    ⁢                  ⁢          (      P      )        =            1      2        ⁢          (                                    1                                1                                0                                0                                                1                                1                                0                                0                                                0                                0                                0                                0                                                0                                0                                0                                0                              )      
An analyzer is a polarizer that is present after the sample in an ellipsometry system. The Mueller matrix describing an Analyzer is identical to that of a polarizer.
      Analyzer    ⁢                  ⁢          (      A      )        =            1      2        ⁢          (                                    1                                1                                0                                0                                                1                                1                                0                                0                                                0                                0                                0                                0                                                0                                0                                0                                0                              )      
Compensators act by retarding one component of the transverse electromagnetic wave with respect to its orthogonal component. This effect is described by the following expression, where the retardance (d) is a function of the extraordinary and ordinary refractive indices (ne, no) of the birefringent crystal and thickness (T) of the material through which the electromagnetic radiation of wavelength (λ) propagates.
  d  =                    2        ⁢                                  ⁢        π            λ        ·                                  n          e                -                  n          o                            ·    T  
The Mueller matrix describing a general compensator is as follows:
      Compensator    ⁡          (      C      )        =      (                            1                          0                          0                          0                                      0                          1                          0                          0                                      0                          0                                      cos            ⁡                          (              d              )                                                            -                          sin              ⁡                              (                d                )                                                                          0                          0                                      sin            ⁡                          (              d              )                                                            cos            ⁡                          (              d              )                                            )  
For any element that has characteristic polarization axes, such as a polarizer or compensator, rotation matrices are used to describe the azimuthal position (θ) of the element with respect to the plane of incidence.
      Rotation    ⁡          (      R      )        =      (                            1                          0                          0                          0                                      0                                      Cos            ⁡                          (                              2                ⁢                                                                  ⁢                θ                            )                                                            Sin            ⁡                          (                              2                ⁢                                                                  ⁢                θ                            )                                                0                                      0                                      -                          Sin              ⁡                              (                                  2                  ⁢                                                                          ⁢                  θ                                )                                                                          Sin            ⁡                          (                              2                ⁢                                                                  ⁢                θ                            )                                                0                                      0                          0                          0                          1                      )  
The most general mathematical description of a sample is a Mueller matrix consisting of 16 elements which can fully describe any change in polarization state of a beam caused by a sample.
      Mueller    ⁢                  ⁢    Matrix    ⁢                  ⁢          Sample      ⁡              (                  S          MM                )              =      (                                        m            11                                                m            12                                                m            13                                                m            14                                                            m            21                                                m            22                                                m            23                                                m            24                                                            m            31                                                m            32                                                m            33                                                m            34                                                            m            41                                                m            42                                                m            43                                                m            44                                )  
In traditional ellipsometry, the polarization change caused by a sample was described by two parameters, ψ and Δ, but this notation is insufficient for describing partially polarized beams or depolarizing samples.
      Isotropic    ⁢                  ⁢          Sample      ⁡              (                  S          Iso                )              =      (                            1                                      -                          Cos              ⁡                              (                                  2                  ⁢                  ψ                                )                                                              0                          0                                                  -                          Cos              ⁡                              (                                  2                  ⁢                                                                          ⁢                  ψ                                )                                                              1                          0                          0                                      0                          0                                                    Sin              ⁡                              (                                  2                  ⁢                  ψ                                )                                      ⁢                          Cos              ⁡                              (                Δ                )                                                                                        Sin              ⁡                              (                                  2                  ⁢                                                                          ⁢                  ψ                                )                                      ⁢                          Sin              ⁡                              (                Δ                )                                                                          0                          0                                                    -                              Sin                ⁡                                  (                                      2                    ⁢                    ψ                                    )                                                      ⁢                          Sin              ⁡                              (                Δ                )                                                                                        Sin              ⁡                              (                                  2                  ⁢                  ψ                                )                                      ⁢                          Cos              ⁡                              (                Δ                )                                                          )  
An alternative notation can be used to fully describe isotropic samples and partially polarized beams. The isotropic quantities are related in the following manner and can be substituted into the previous matrix as shown:
      N    =          cos      ⁡              (                  2          ⁢          ψ                )                  C    =                  sin        ⁡                  (                      2            ⁢            ψ                    )                    ⁢              cos        ⁡                  (          Δ          )                          S    =                  sin        ⁡                  (                      2            ⁢            ψ                    )                    ⁢              sin        ⁡                  (          Δ          )                                Isotropic      ⁢                          ⁢              Sample        ⁡                  (                      S            Iso                    )                      =                  (                                            1                                                      -                N                                                    0                                      0                                                                          -                N                                                    1                                      0                                      0                                                          0                                      0                                      C                                      S                                                          0                                      0                                                      -                S                                                    C                                      )            .      
The N, C, & S notation is advantageous because it provides a simple relationship to depolarization. Depolarization is defined as the transformation of fully polarized electromagnetic radiation to partially polarized electromagnetic radiation and can be expressed in the following manner for isotropic samples:% Depolarization=100%·(1−N2−C2−S2)
Depolarization can be caused by a variety of factors, including; surface electromagnetic radiation scattering, sample non-uniformity, spectrometer bandwidth resolution, angular spread from a non-collimated input beam, and incoherent summation of electromagnetic radiation reflecting from the backside of a substrate. Depolarization measurements help to identify non-idealities in the sample or system.
One of the most common elements used in prior art snapshot ellipsometers is a wedge of some birefringent material. Because the retardance of a compensator is proportional to its thickness, a birefringent optic with spatially variable thickness has different values of retardance at different positions. Because different portions of a measurement beam interact with different portions of the optic, the polarization state of the measurement beam becomes spatially modulated.
The simplest example of such an optic is a linear wedge made of a birefringent crystal. The thickness T of such a wedge can be described as a function of spatial position x along the direction of variation, where the rate of change is defined by the slope of the wedge, w.T(x)=w·x 
Substituting the wedge thickness into the general retardance equation, the retardance across the wedge can be described as a function of spatial position.
      d    ⁡          (      x      )        =                    2        ⁢                                  ⁢        π            λ        ·                                  n          e                -                  n          o                            ·    w    ·    x  
A new term D is defined to express the rate of retardance change.
      D    =                            2          ⁢                                          ⁢          π                λ            ·                                            n            e                    -                      n            o                                      ·      w                  d      ⁡              (        x        )              =          D      ·      x      
The Mueller matrix description of a wedged birefringent crystal is like that of a standard compensator, where the spatial retardance is defined by d(x).
      Wedge    ⁡          (              W        ⁡                  [          x          ]                    )        =      (                            1                          0                          0                          0                                      0                          1                          0                          0                                      0                          0                                      cos            ⁡                          (              Dx              )                                                            -                          sin              ⁡                              (                Dx                )                                                                          0                          0                                      sin            ⁡                          (              Dx              )                                                            cos            ⁡                          (              Dx              )                                            )  
Any optic with birefringence and variable thickness will spatially modulate the polarization state of a beam incident thereon, and the term ‘wedge’ is used herein to describe any optic with such a thickness variation, not to specifically define that the thickness varies linearly or continuously as in the example above. An optic exhibiting a discrete stepwise thickness profile or one that varies nonlinearly could be easily substituted by one skilled in the art.
A common modification of a linearly wedged retarder as described above is a Babinet compensator. A Babinet compensator is a set of two equally-wedged crystals of uniaxially anisotropic material. The wedges are oriented with the two wedged faces in contact or with a small gap between them and the two opposite faces parallel to each other and normal to the incident beam, as illustrated in FIG. 3B. The optic axes of the two wedges are orthogonal to each other and the beam.
The Mueller matrix describing the Babinet compensator can be defined by matrix multiplication of the two component wedges with optic axes of W1, W2:Babinet[x]=R(−W2)·W2[x]·R(W2)·R(−W1)·W1[x]·R(W1)
Because the slopes of the two wedges are equal and opposite, the retardance rate is defined as B for the first wedge and −B for the second.
      Babinet    ⁡          [      x      ]        =      (                            1                          0                          0                          0                                      0                          1                          0                          0                                      0                          0                                      Cos            ⁡                          [                              2                ⁢                Bx                            ]                                                            -                          Sin              ⁡                              [                                  2                  ⁢                  Bx                                ]                                                                          0                          0                                      Sin            ⁡                          [                              2                ⁢                Bx                            ]                                                            Cos            ⁡                          [                              2                ⁢                Bx                            ]                                            )  
This matrix is clearly equivalent to that of a single wedge with double the rate of retardance variation than the component wedges. Recognizing that a Babinet compensator therefore produces the same polarization modulation as a single wedge with a different slope, it should be obvious to one skilled in the art that birefringent wedges can be replaced with combinations of wedges in order to produce spatial modulation of polarization state within a beam. The benefit of a Babinet compensator is that the retardance near the center of the optic is zero-order and the beam experiences less deviation and separation when transmitted therethrough. As such, the term ‘wedge’ is further understood to refer both to individual wedges and combinations of wedges or optics, such as Babinet compensators, which are used to affect a spatially varying retardance to a beam along one azimuth.
U.S. Pat. No. 6,052,188A (1998) by Fluckiger et al. describes an ellipsometer that uses a single wedge in order to impart spatially varied retardance to a beam. The ellipsometer system described by Fluckiger was novel in its implementation but limited in its inability to detect all sample parameters describing isotropic samples and severely limited in measuring anisotropic samples. Fluckiger specified a Babinet compensator to be preferred over a single wedge for the reasons given above.
Using the matrices defined above, the spatially varying signal intensity of a single wedge system can be expressed by the following matrix multiplication:Intensity(x)=D·R(−A)·A·R(A)·R(−W)·W[x]·R(W)·S·R(−P)·P·R(P)·L 
The polarizer and analyzer should be oriented at a non-eigenpolarization state of the wedge for each measurement. Assuming an isotropic sample and that polarizer and analyzer azimuth are set to 45*, the spatial variation of beam intensity can be expressed as follows:Intensity(x)=1+C·Cos[2·D·x]·S·Sin[2·D·x]
A Fourier transformation of the signal along the axis of variation on the detector decomposes the expression into components occurring at different spatial frequencies.
                              α          k                =                  Re          ⁢                      {                                          1                P                            ⁢                                                ∫                                                            -                      P                                        2                                                        P                    2                                                  ⁢                                                                            (                                              1                        +                                                  C                          ·                                                      Cos                            ⁡                                                          [                                                              2                                ·                                D                                ·                                x                                                            ]                                                                                                      -                                                  S                          ·                                                      Sin                            ⁡                                                          [                                                              2                                ·                                D                                ·                                x                                                            ]                                                                                                                          )                                        ·                                          e                                                                        -                          2                                                ⁢                        π                        ⁢                                                                                                  ⁢                                                  i                          ⁡                                                      (                                                          k                              P                                                        )                                                                          ⁢                        x                                                                              ⁢                                                                          ⁢                  dx                                                      }                                                            β          k                =                  I          ⁢                                          ⁢          m          ⁢                      {                                          1                P                            ⁢                                                ∫                                                            -                      P                                        2                                                        P                    2                                                  ⁢                                                                            (                                              1                        +                                                  C                          ·                                                      Cos                            ⁡                                                          [                                                              2                                ·                                D                                ·                                x                                                            ]                                                                                                      -                                                  S                          ·                                                      Sin                            ⁡                                                          [                                                              2                                ·                                D                                ·                                x                                                            ]                                                                                                                          )                                        ·                                          e                                                                        -                          2                                                ⁢                        π                        ⁢                                                                                                  ⁢                                                  i                          ⁡                                                      (                                                          k                              P                                                        )                                                                          ⁢                        x                                                                              ⁢                                                                          ⁢                  dx                                                      }                              
The real (αk) and imaginary (βk) Fourier coefficients are non-zero at certain frequencies, k, related to the spatial variation of the compensator. Fourier transformation of the theoretical intensity expression identifies a single harmonic frequency (k=2D) and a DC (k=0) term. The theoretical Fourier coefficients are related to sample parameters as follows:α0=1α2D=C β2D=−S 
Using the theoretical expressions for Fourier coefficients to solve for sample parameters, it's evident the snapshot ellipsometer using a single wedge can measure only two sample parameters in any single measurement, as shown in the following Mueller matrix description of the sample (X indicates insensitivity to a parameter).
      Sample    ⁢                  ⁢          (      S      )        =      (                            1                          X                          0                          X                                      X                          X                          X                          X                                      0                          X                                      α                          2              ⁢              D                                                X                                      0                          X                                      -                          β                              2                ⁢                D                                                              X                      )  
A single wedge system therefore cannot fully characterize an isotropic sample because only two of the three sample parameters can be measured simultaneously. As arranged, systematic errors and noise in the measured data are amplified when measuring samples with Psi near 45°. For a single wedge system, all arrangements of polarizing optics result in amplified error in Psi and Delta for certain sample types.